Question
Which of the following statement is true ?
A.
The point $$A\left( {0,\, - 1} \right),\,B\left( {2,\,1} \right),\,C\left( {0,\,3} \right)$$ and $$D\left( { - 2,\,1} \right)$$ are vertices of a rhombus.
B.
The points $$A\left( { - 4,\, - 1} \right),\,B\left( { - 2,\, - 4} \right),\,C\left( {4,\,0} \right)$$ and $$D\left( {2,\,3} \right)$$ are vertices of a square.
C.
The points $$A\left( { - 2,\, - 1} \right),\,B\left( {1,\,0} \right),\,C\left( {4,\,3} \right)$$ and $$D\left( {1,\,2} \right)$$ are vertices of a parallelogram.
D.
None of these
Answer :
The points $$A\left( { - 2,\, - 1} \right),\,B\left( {1,\,0} \right),\,C\left( {4,\,3} \right)$$ and $$D\left( {1,\,2} \right)$$ are vertices of a parallelogram.
Solution :
$$\left( {\bf{A}} \right)$$ Here $$A\left( {0,\, - 1} \right),\,B\left( {2,\,1} \right),\,C\left( {0,\,3} \right),\,D\left( { - 2,\,1} \right)$$
For a rhombus all four sides are equal but the diagonal are not equal, we see
$$AC = \sqrt {0 + {4^2}} = 4,\,\,BD = \sqrt {{4^2} - 0} = 4$$
Since diagonals are equals therefore it is a square, not rhombus.
$$\left( {\bf{B}} \right)$$ Here $$AB = \sqrt {{2^2} + {{\left( { - 3} \right)}^2}} = \sqrt {13} ,\,\,BC = \sqrt {{6^2} + {4^2}} = \sqrt {52} $$
Since $$AB \ne BC$$ therefore it is not square.
$$\left( {\bf{C}} \right)$$ In this case mid point of $$AC$$ is $$\left( {\frac{{4 - 2}}{2},\,\frac{{3 - 1}}{2}} \right){\text{ or }}\left( {1,\,1} \right)$$
Also mid-point of diagonal $$BD\left( {\frac{{1 + 1}}{2},\,\frac{{0 + 2}}{2}} \right){\text{ or }}\left( {1,\,1} \right)$$
Hence the points are vertices of a parallelogram.